MHB Hyperreal Intermediate Value Theorem

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The discussion centers on the application of the Intermediate Value Theorem (IVT) to hyperreal numbers. A participant questions whether the IVT holds for continuous functions defined on hyperreal intervals, specifically when considering closed intervals that may be unbounded. They reference Jerome Keisler's book, which provides a proof of the IVT in the context of hyperreals. The conversation also touches on the limitations of using Los's theorem for proving this property. Overall, the inquiry seeks to clarify the relationship between continuity in hyperreal functions and the IVT.
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I haven't thought too much about this, but it seems to me that the intermediate value theorem would transfer. Am I incorrect?
 
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conscipost said:
I haven't thought too much about this, but it seems to me that the intermediate value theorem would transfer. Am I incorrect?

In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this.
The chapters and whole book is a free down-load HERE.
 
Plato said:
In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this.
The chapters and whole book is a free down-load HERE.

This was not quite what I was thinking about. But I think it helps clarify the question I had. I have not thought much about the topology of the hyperreals but what I'm asking is:

Say I have a continuous function defined on the hyperreals and I take a "closed" interval [a,b], that is the set of all hyperreals between a and b, where a and b may very well be unlimited, does it follow that for any c: f(a)<c<f(b) there exists an x so that f(x)=c.

The star transform of [a,b] wouldn't give unlimited elements as far I'm looking into it, so I'm afraid this would have to be proven using something other than Los's theorem.
 
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Thread 'Problem with calculating projections of curl using rotation of contour'

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